Median Ratings

Median Ratings or Majoritarian Score is a voting system used for single-seat elections. It is like Score voting except that the candidate with the highest median score, rather than the highest sum score, is elected.

Contents

Example

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.

The candidates for the capital are:

Memphis, the state's largest city, with 42% of the voters, but located far from the other cities

Suppose that voters were told to grant 1 to 4 points to each city such that their most liked choice(s) got 4 points, and least liked choice(s) got 1 point.

City

Memphis

Nashville

Chattanooga

Knoxville

"4" votes

42

26

15

17

"3" votes

0

42

43

15

"2" votes

0

32

42

26

"1" votes

58

0

0

42

Median

1

3

3

2

There is a tie between Nashville and Chattanooga.

Advantages

The main advantages of using the median instead of the sum or arithmetic mean of the individual ratings are:

The result is determined by majorities: when a majority specifies a ranking at least a, then the median is also at least a, and when a majority specifies a ranking at most a, then the median is also at most a.

Disadvantages

The example illustrates one of the major flaws of Median Ratings. Because the median, by definition, is either the value of a single rating or the average of two ratings, there are very few possible values for a median score, and this makes ties much more likely than in Score voting.

Given that a big range like 0-999 is probably an awkward solution, because nobody senses so sophisticated differences between candidates, a shorter range like 0-9 and a resolving method for ties might work better. Ties could be solved by using the average. Or by using the interquartile mean. One could also select just enough ratings above and under the median that the average of those gives a clear front runner. But that would fail independence of irrelevant alternatives. Yet another solution would be to take the average of the rating above and under each median in a tie. If there is still a tie, we take the average of the rating above the rating above and the rating under the rating under each median that is still in a tie. And if there is still a tie, we take the rating above the rating above the rating above an the rating under the rating under the rating under each median that is still in a tie etc.

Another problem with Median Ratings is its failure of the Blank Ballot Criterion. For example, with the ballots

A=1, B=4

A=2, B=4

A=8, B=4

A=9, B=4

A (median rating=5) beats B (median rating=4), but if the ballot (A=0, B=0) is added, then A's median rating drops to 2, causing B to win.